Composition operators on weighted Bergman spaces of Dirichlet series

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Composition operators on weighted Bergman spaces of Dirichlet series

Maxime Bailleul

To cite this version:

Maxime Bailleul. Composition operators on weighted Bergman spaces of Dirichlet series. 2014. �hal-

00938088�

Composition operators on weighted Bergman spaces of Dirichlet series

Maxime Bailleul

Abstract

We study boundedness and compactness of composition operators on weighted Bergman spaces of Dirichlet series. Particularly, we obtain in some specic cases, upper and lower bounds of the essential norm of these operators and a criterion of compactness on classicals weighted Bergman spaces. Moreover, a sucient condition of compactness is obtained using the notion of Carleson's measure.

Mathematics Subject Classication.

Primary: 47B33 - Secondary: 46E15.

Keywords.

weighted Bergman spaces - Carleson measure - composition operator - Dirichlet series - essential norm - generalized Nevanlinna counting function

1 Introduction

In [10], the authors dened the Hardy space H ² of Dirichlet series with square-summable coecients. Thanks to the Cauchy-Schwarz inequality, it is easy to see that H ² is a space of analytic functions on C

:= {s ∈ C , <(s) > ¹ ₂ } and that this domain is maximal. F. Bayart introduced in [2] the more general class of Hardy spaces of Dirichlet series H ^p ( 1 ≤ p < +∞ ).

In [9], the bounded composition operators on H ² are characterized, that is to say the holomorphic functions Φ : C

→ C

such that for any f ∈ H ² , f ◦ Φ ∈ H ² . We denote by D the class of functions f which admit representation by a convergent Dirichlet series in some half-plane and for θ ∈ R, C θ will be the following half-plane {s ∈ C , <(s) > θ} . We shall denote C + instead of C 0 . Theorem ([9],[18]). The function Φ determines a bounded composition operator on H ² if and only:

Φ(s) = c 0 s + ϕ(s)

where c ₀ is a nonnegative integer, ϕ ∈ D and ϕ converges uniformly in C ε for every ε > 0 and has the following properties:

a) If c ₀ ≥ 1 , ϕ( C + ) ⊂ C + . b) If c ₀ = 0 , ϕ( C + ) ⊂ C 1/2 .

Moreover, when c ₀ ≥ 1 , C _Φ is a contraction.

This theorem is a strengthening of the original result (see [18],Th3.1). In

[2], F. Bayart proved that this result is true on the space H ^p when p ≥ 1 and

c 0 6= 0 . When c 0 = 0 the condition is necessary for every p ≥ 1 and sucient

if p is even. The goal of this paper is to study composition operators on the

weighted Bergman A ^p _µ spaces of Dirichlet series dened in [1]. In order to dene

them, we need to recall the Bohr's point of view principle and the denition of

the Hardy spaces H ^p .

Let n ≥ 2 be an integer, we shall denote e _n the function dened by e _n (z) = n ^−z . The integer n can be written (in a unique way) as a product of prime numbers n = p ^α ₁

· · p ^α _k

where p 1 = 2, p 2 = 3 etc . . . If s is a com- plex number, we consider z = (p ^−s ₁ , p ^−s ₂ , . . . ) . let f be a Dirichlet series of the following form

f =

+∞

X

n=1

a n e n , (1)

then

f (s) =

+∞

X

n=1

a n (p ^−s ₁ ) ^α

· · (p ^−s _k ) ^α

=

+∞

X

n=1

a n z ₁ ^α

· · z _k ^α

.

So we can see a Dirichlet series as a Fourier series on the innite-dimensional polytorus T ^∞ = {(z 1 , z 2 , · · · ), |z i | = 1, ∀i ≥ 1} . We shall denote this Fourier series D(f ) .

Let us x now p ≥ 1 , the space H ^p ( T ^∞ ) is the closure of the set of analytic polynomials with respect to the norm of L ^p ( T ^∞ , m) where m is the normalized Haar measure on T ^∞ (see [6] for more details). Let f be a Dirichlet polyno- mial, by the Bohr's point of view D(f ) is an analytic polynomial on T ^∞ and by denition kf k _H

:= kD(f )k _H

(

) . The space H ^p is dened by taking the closure of Dirichlet polynomials with respect to this norm. Consequently H ^p and H ^p ( T ^∞ ) are isometrically isomorphic.

For σ > 0 , f σ will be the translate of f by σ , ie f σ (s) := f (σ + s) . We shall denote by P the set of Dirichlet polynomials.

Let p ≥ 1 , P ∈ P and let µ be a probability measure on (0, +∞) such that 0 ∈ Supp(µ) . Then

kP k _A

:=

Z +∞

0

kP σ k ^p _H

dµ(σ) 1/p

.

A ^p _µ is the completion of P with respect to this norm. When dµ(σ) = 2e ^−2σ dσ , these spaces are simply denoted by A ^p . It is shown in [1] that they are spaces of convergent Dirichet series on C 1/2 .

In section 2 , we recall some facts about Dirichlet series and precise some notations.

In section 3 , we study the boundedness of C Φ on A ^p _µ . For Φ(s) = c 0 s + ϕ(s) with c 0 ≥ 1 and ϕ ∈ D , Φ induces a bounded composition operator from A ^p _µ to A ^p _µ if and only if ϕ converges uniformly on C ε for every ε > 0 and ϕ( C + ) ⊂ C + . When c 0 = 0 , we give sucient and necessary conditions.

In section 4 , we dene generalized Nevanlinna counting function in order to obtain some estimates of the essential norm of C Φ on H ² and A ² _µ . Particularly we obtain a criterion of compactness of C Φ in some specic cases.

In section 5 , we obtain a majorization of the generalized Nevanlinna counting

function with help of the associated Carleson function (see denition in section

5 ) and then we get other sucient conditions of compactness of C _Φ .

2 Background material

Let f be a Dirichlet series of form (1) . We do not recall the denition of abscissa of simple (resp. absolute) convergence denoted by σ c (resp. σ a ), see [17] or [21] for more details. We shall need the two other following abscissas:

σ _u (f ) = inf{a | The series (1) is uniformly convergent for <(s) > a}

= abscissa of uniform convergence of f.

σ b (f ) = inf{a | the function f has an analytic, bounded extension for <(s) > a}

= abscissa of boundedness of f.

It is easy to see that σ c (f ) ≤ σ u (f ) ≤ σ a (f ) . An important result is that σ u (f ) and σ b (f ) coincide: this is the Bohr's theorem (see [4]), it is really impor- tant for the study of H ^∞ , the algebra of bounded Dirichlet series on the right half-plane C + (see [15]). We shall denote by k · k ∞ the norm on this space:

kf k _∞ := sup

<(s)>0

|f (s)|.

The innite-dimensional polytorus T ^∞ can be identied with the group of complex-valued characters χ on the positive integers which satisfy the following properties

|χ(n)| = 1 ∀n ≥ 1,

χ(nm) = χ(n) χ(m) ∀n, m ≥ 1.

To obtain this identication for χ = (χ 1 , χ 2 , . . . ) ∈ T ^∞ , it suces to dene χ on the prime numbers by χ(p _i ) = χ _i and use multiplicativity.

Let f be a Dirichlet series absolutely convergent in a half-plane. For any sequences (τ _n ) ⊂ R, we consider vertical translations of f :

(f τ

(s)) := (f (s + iτ n )).

By Montel's theorem, this sequence is a normal family in the half-plane of absolute convergence of f and so there exists a convergent subsequence, say (f τ

) , such that f τ

converges uniformly on compact subsets of the domain of absolute convergence of f to a limit function f ˜ . We say that f ˜ is a vertical limit of f .

Proposition ([10]). Let f be a Dirichlet series of the form (1), absolutely con- vergent in a half-plane. The vertical limit functions of f are exactly the functions of the form f χ = X

n≥1

a n χ(n) e n where χ ∈ T ^∞ .

When f ∈ A ^p _µ , the vertical limits have good properties:

Proposition ([1]). Let p ≥ 1 and f ∈ A ^p _µ . For almost all χ ∈ T ^∞ , f _χ converges on C + .

Let Φ(s) = c ₀ s + ϕ(s) where c ₀ is a nonnegative integer and ϕ ∈ D . For τ ∈ R, we dene Φ τ by Φ τ (s) = c 0 s + ϕ(s + iτ ) , then if χ ∈ T ^∞ the function Φ χ dened by

Φ χ (s) = c 0 s + ϕ χ (s)

is a vertical limit of the functions (Φ _τ ) _τ∈

.

In section 4 , we shall consider the Hilbert spaces of Dirichlet series H ² and A ² _µ where µ is a probability measure on (0, +∞) such that 0 ∈ Supp(µ) . Moreover we shall assume that dµ(σ) = h(σ)dσ where h is a nonnegative function such that khk _L

((0,+∞)) = 1 . We see easily that for an element of A ² _µ with form (1) :

kf k _A

=

+∞

X

n=1

|a n | ² w h (n) 1/2

where

w h (n) = Z +∞

0

n ^−2σ h(σ)dσ ∀n ≥ 1.

Examples. We follow notations from [1]: let α > −1 , we denote µ _α the prob- ability measure dened on (0, +∞) by

dµ α (σ) = 2 ^α+1

Γ(α + 1) σ ^α exp(−2σ)dσ.

We denote this space A ² _α instead of A ² _µ

and the corresponding weight by (w ^α _n ) . Then for every n ≥ 1 ,

w ^α _n = 1

(log(n) + 1) ^α+1 .

We shall denote (e ^µ _n ) n≥1 the orthonormal basis of A ² _µ dened by e ^µ _n := e _n

p w h (n) .

3 Boundedness of composition operators

In the sequel µ will be a probability measure on (0, +∞) such that 0 ∈ Supp(µ) (as in [1]).

Theorem 1. Let Φ : C

→ C

be an analytic function of the form Φ(s) = c ₀ s + ϕ(s) where c ₀ ≥ 1 and ϕ ∈ D . Then C _Φ is bounded on A ^p _µ if and only ϕ converges uniformly in C ε for every ε > 0 and ϕ( C + ) ⊂ C + and in this case C _Φ is a contraction.

Remark. The condition Φ(s) = c 0 s + ϕ(s) where c 0 is a nonnegative integer is an arithmetical condition: Φ has this form if and only if P ◦ Φ ∈ D for every P ∈ P (see [9],Th.A).

Proof.

B Assume that ϕ converges uniformly on C ε for every ε > 0 and that ϕ( C + ) ⊂ C + . Let P be a Dirichlet polynomial, P ◦ Φ is then a Dirichlet series which is bounded on C + (because Φ( C + ) ⊂ C + ) and so P ◦ Φ belongs to H ^∞ and consequently to H ^p . By [1],Th.6(ii) we know that

kP ◦ Φk ^p _A

=

Z +∞

0

k(P ◦ Φ) σ k ^p _H

dµ(σ).

Now, the key point is to use the boundedness of the composition operators on H ^p . We remark that for each σ > 0 ,

(P ◦ Φ) σ (it) = P (Φ(σ + it)) = P σ (Φ(σ + it) − σ).

Let Φ e dened on C + by Φ(s) = Φ(σ e + s) − σ . We claim that Φ e veries the condition of the boundedness of C

Φ

on H ^p : it can be written Φ(s) = e c 0 s + ϕ(s) e where ϕ e converges uniformly on C ε for every ε > 0 . Then it suces to check that ϕ( e C + ) ⊂ C + but ϕ(s) = e ϕ(s + σ) + (c 0 − 1)σ for every s ∈ C + and the conclusion is clear because c ₀ ≥ 1 and ϕ( C + ) ⊂ C + .

Now we apply the result on the boundedness on H ^p . Since c ₀ ≥ 1 , C

Φ

is a contraction on H ^p and we obtain ( σ is xed):

k(P ◦ Φ) _σ k ^p _H

= kP σ (e Φ)k ^p _H

≤ kP σ k ^p _H

. This is true for every σ > 0 , so

Z +∞

0

k(P ◦ Φ) σ k ^p _H

dµ(σ) ≤ Z +∞

0

kP σ k ^p _H

dµ(σ).

Hence kP ◦ Φk _A

≤ kP k _A

. Now, by density of P in A ^p _µ , C Φ : P ∩ A ^p _µ → A ^p _µ extends to a bounded operator T : A ^p _µ → A ^p _µ such that T (P ) = P ◦ Φ for every P ∈ P . Let f ∈ A ^p _µ , there exists a sequence (P _n ) ⊂ P such that (P _n ) converges to f in norm, then by continuity of the point evaluation at s ∈ C 1/2 on A ^p _µ (see [1],Th.1):

|T (f)(s) − T (P n )(s)| ≤ kδ s k _(A

₎

kT(f ) − T (P n )k _A

≤ kδ s k _(A

₎

kT k kf − P n k _A

and then T (P n )(s) = P n (Φ(s)) converges to T (f )(s) when n goes to inn- ity. But by evaluation at Φ(s) ∈ C 1/2 , P n (Φ(s)) converges to f (Φ(s)) , so T (f )(s) = f (Φ(s)) = C Φ (f )(s) and the result is proved.

B Now assume that Φ induces a bounded composition operator on A ^p _µ . The result follows from ideas similar to the ones of the proof of Th.B from [9]. We only give a sketch of the proof.

Let f ∈ A ^p _µ . For every χ ∈ T ^∞ , we can show that (see [9], Prop4.3) for every s ∈ C 1/2 ,

(f ◦ Φ) _χ (s) = f _χ

◦ Φ _χ (s). (∆)

Since f and f ◦ Φ belong to A ^p _µ , (f ◦ Φ) _χ and f _χ

extend analytically on C +

almost-surely relatively to χ . As in Prop5.1 from [9], we can show that Φ _χ extends analytically on C + for every χ ∈ T ^∞ .

Now, let χ ∈ T ^∞ such that f _χ

, Φ _χ and (f ◦ Φ) _χ extend on C + .

If Φ _χ does not map C + to C + , then there exists s ₀ ∈ C + such that Φ _χ (s ₀ ) is in the imaginary axis and with an argument of connectedness, we can choose it such that Φ

_χ (s 0 ) 6= 0 . Then, by (∆) , f χ

has an analytic extension on a small segment on the imaginary axis near Φ χ (s 0 ) . We denote (p n ) _n≥1 the sequence of prime numbers. In [2] it is shown that

f = X

n≥1

e p

√ p _n log(p _n ) ∈ H ^p

and that almost surely relatively to χ , f _χ

does not extend analytically to any region larger than C + . But f ∈ A ^p _µ because H ^p ⊂ A ^p _µ . Consequently Φ _χ has to map C ⁺ to C ⁺ almost surely relatively to χ and then by Proposition 4.1 from [9], Φ( C ⁺ ) ⊂ C ⁺ . Finally, it suces to apply Th3.1 from [18] to obtain that ϕ( C + ) ⊂ C + and ϕ converges uniformly on C ε for all ε > 0 .

Remark. In the previous proof we used that Φ _u − u : C + → C + when c ₀ ≥ 1 . We can see this result as an equivalent of the Schwarz's lemma in this framework.

When c ₀ = 0 , we obtain the following result.

Theorem 2. Let p ≥ 1 and Φ : C

→ C

be an analytic function belonging to D . Then

(i) If C Φ is bounded on A ^p _µ then Φ converges uniformly on C ε for every ε > 0 and Φ( C + ) ⊂ C 1/2 .

(ii) If Φ converges uniformly on C ε for every ε > 0 and Φ( C + ) ⊂ C 1/2+η

(where η > 0 ) then C _Φ is Hilbert-Schmidt on A ² _µ , a fortiori bounded.

(iii) If C Φ is bounded on A ² _µ , then it is bounded on A ^2k _µ for any k ≥ 1 . Conse- quently C Φ is bounded on A ^2k _µ for any k ≥ 1 under conditions of (ii) . Proof.

(i) As in the case c 0 ≥ 1 , the proof follows from [9], Th.B and some easy adaptations.

(ii) By Lemma 5 from [2] we have:

kn ^−Φ

k ² _H

= Z

T^∞

Z 1 0

|n ^−Φ

^(σ+it) | ² dtdm(χ) ∀n ≥ 1.

Now Φ : C + → C 1/2+η and then by Proposition 4.1 from [9], the same holds for Φ χ for every χ ∈ T ^∞ . Consequently

kn ^−Φ

k ² _H

≤ n ^−1−2η ∀n ≥ 1.

Now we show that C Φ is Hilbert-Schmidt and then obviously bounded. The sequence (e ^µ _n ) _n≥1 is an orthonormal basis of A ² _µ and C Φ (e ^µ _n ) ∈ H ^∞ for every n ≥ 1 : indeed e ^µ _n ∈ P and Φ( C + ) ⊂ C + . Then by [1], Th.6(ii):

+∞

X

n=1

kC _Φ (e ^µ _n )k ² _A

=

+∞

X

n=1

Z +∞

0

k(C _Φ (e ^µ _n )) _σ k ² _H

dµ(σ)

=

+∞

X

n=1

Z +∞

0

kn ^−Φ

k ² _H

w _h (n) dµ(σ) ≤

+∞

X

n=1

Z +∞

0

n ^−1−2η w _h (n) dµ(σ)

=

+∞

X

n=1

n ^−1−2η w _h (n) .

Now in [16], it is shown that the weight (w h (n)) decreases more slowly than any negative power of n so there exists C > 0 such that w h (n) ≥ Cn ^−η for every n ≥ 1 , then

+∞

X

n=1

kC Φ (e ^µ _n )k ² _A

≤ 1

C

+∞

X

n=1

1

n ^1+η < +∞

and so C _Φ is Hilbert-Schmidt.

(iii) Assume that C _Φ is bounded on A ² _µ . Let P be a Dirichlet polynomial, we get

kP ◦ Φk _A

µ

= kP ^k ◦ Φk ^1/k _A

µ

≤ kC Φ k ^1/k kP ^k k ^1/k _A

µ

= kC Φ k ^1/k kP k _A

. By density of the polynomials and boundedness of the point evaluation we obtain the result (as in the proof of Theorem 1).

Questions.

Is it true that C Φ is actually nuclear?

Can we choose η = 0 in (ii) to get boundedness?

4 Compactness and Nevanlinna counting function

Let p ≥ 1 and X = H ^p or A ^p _µ . We begin this section with the following criterion of compactness:

Proposition 1. Let Φ : C 1/2 → C 1/2 be an analytic function which induces a bounded composition operator on X . Then C Φ is compact on X if and only if for every bounded sequence (f n ) in X which converges to 0 uniformly on C 1/2+ε

for every ε > 0 , one has kC Φ (f n )k X → 0 .

Proof. The point-evaluation at s ∈ C 1/2 is bounded on X , X has the Fatou- property (in the sense of [13], Prop 4.8) and there exists a Montel's theorem for H ^∞ (see [3]). Then it suces to apply the same reasoning than in the proof of Proposition 4.8 from [13].

Remark. With this criterion, it is easy to see that if C _Φ is compact on H ² (resp. A ² _µ ) then C _Φ is compact on H ^2k (resp. A ^2k _µ ) for any k ≥ 1 .

A Sucient conditions of compactness

In this section, we follow ideas from [3]: we dene the generalized Nevanlinna counting function associated to the spaces A ² _µ in order to study the compactness of composition operators on A ² _µ . We need some preliminaries from [1].

For σ > 0 , we dene β h (σ) :=

Z σ 0

(σ − u)h(u) du = Z σ

0

Z t 0

h(u) dudt.

We point out that if h is continuous, the two rst derivatives of β h are β _h ⁰ (σ) = R σ

0 h(u)du, β _h ⁰⁰ (σ) = h(σ).

Remark. If we denote β _α instead of β _h for the spaces A ² _α , it is easy to see that

β _α (σ) ≈ σ ^α+2 when σ is close to 0 .

Theorem ("Littlewood-Paley formula" [1]). Let η be a Borel probability mea- sure on R. Then

kf k ² _A

µ

= |f (+∞)| ² + 4 Z

T^∞

Z +∞

0

Z

R

β h (σ)|f _χ

(σ + it)| ² dη(t)dσdm(χ).

Remark. For a Dirichlet series f with expansion (1) , f (+∞) stands for a ₁ , the constant coecient.

Notation. We shall call a c 0 -symbol any analytic function Φ : C 1/2 → C 1/2 of the form Φ(s) = c 0 s + ϕ(s) where c 0 is a nonnegative integer and ϕ ∈ D such that C Φ is bounded on A ² _µ . When c 0 ≥ 1 it is equivalent to the fact that ϕ converges uniformly on C ε for every ε > 0 and ϕ( C + ) ⊂ C + . We point out that in the sequel, the hypothesis " c 0 ≥ 1 " will be crucial.

Denition 1. We dene the generalized Nevanlinna counting function associ- ated to a function β : (0, +∞) → (0, +∞) by

N β,Φ (s) =



 

  X

a∈C

Φ(a)=s

β(<(a)) if s ∈ Φ( C + ),

0 else.

When β(σ) = σ , we just denote N Φ this function.

Remarks.

(i) N Φ is associated to the space H ² , it was already dened in [3].

(ii) N β,Φ is the equivalent in our framework of Dirichlet series of the generalized Nevanlinna counting function dened in [11] in the framework of analytic functions on the unit disk.

Lemma 1. Let Φ be a c 0 -symbol with c 0 ≥ 1 . For every s ∈ C + , we have N _β

_,Φ (s) =

Z <(s) 0

N _Φ

(s)h(u)du where Φ u is dened by Φ u (s) := Φ(s + u) .

Proof. Let s ∈ C + .

N β

,Φ (s) = X

a∈

Φ(a)=s

β h (<(a))

= X

a∈

Φ(a)=s

Z <(a) 0

(<(a) − u)h(u)du

= X

<(a)≤<(s) Φ(a)=s

Z <(a) 0

(<(a) − u)h(u)du

thanks to the "Schwarz's lemma": indeed Φ( C <(a) ) ⊂ C <(a) and then <(a) >

<(s) implies Φ(a) 6= s . Now by the Fubini's theorem:

N _β

_,Φ (s) = Z <(s)

0

X

u<<(a)≤<(s) Φ(a)=s

(<(a) − u)h(u) du.

Now it suces to remark that X

u<<(a)≤<(s) Φ(a)=s

(<(a) − u) = X

u<<(a) Φ

(a−u)=s

(<(a) − u)

= X

0<<(a

) Φ

(a

)=s

<(a ⁰ )

= N Φ

(s).

Proposition 2. Let Φ be a c 0 -symbol with c 0 ≥ 1 . For every s ∈ C + , we have N β

,Φ (s) ≤ β h (<(s))

c 0

. Proof. By the previous lemma, we know that

N β

,Φ (s) = Z <(s)

0

N Φ

(s)h(u)du.

We already pointed out that Φ _u − u : C + → C + and then by Littlewood's inequality ([3],Prop.3):

N Φ

−u (s ⁰ ) ≤ <(s ⁰ ) c 0

∀s ⁰ ∈ C + . But N Φ

(s) = N Φ

−u (s − u) when <(s) > u > 0 . Finally

N β

,Φ (s) ≤ Z <(s)

0

(<(s) − u)h(u) c 0

du = β _h (<(s)) c 0

.

In order to obtain an upper bound for the essential norm of C Φ , we need to obtain some estimates on N β

,Φ . We follow ideas from [3].

Proposition 3. Let Φ : C + → C + and Φ _k : C + → C + where k ≥ 0 , some analytic functions. Assume that (Φ _k ) _k≥0 converges uniformly to Φ on every compact subset of C + , then for every s ∈ C + ,

N β

,Φ (s) ≤ lim inf

k→+∞ N β

,Φ

(s).

Proof. Let us x s ∈ Φ( C + ) (if s / ∈ Φ( C + ) there is nothing to prove) and ε > 0 . Let a ₁ , . . . , a _n ∈ C + satisfying Φ(a _i ) = s for every i = 1, . . . , n and δ > 0 such that 2nδ < ε . (Φ _k ) _k≥0 converges uniformly to Φ on every compact set of C +

and so particularly on each disk D(a i , δ) of center a i and radius δ . Then by the Hurwitz's lemma, there exists K ≥ 0 such that for any k ≥ K , s ∈ Φ k (D(a i , δ)) for every i = 1, . . . , n .

Now let us x k ≥ K , there exist a ^k ₁ ∈ D(a 1 , δ) , ... , a ^k _n ∈ D(a n , δ) such that

Φ(a ^k _i ) = s for every i = 1, . . . , n . By denition of β h , we have

β h (<(a ^k _i )) =

Z <(a

) 0

(<(a ^k _i ) − u)h(u)du

≥

Z <(a

)−δ 0

(<(a i ) − δ − u)h(u)du

= β h (<(a i )) −

δ

Z <(a

)−δ 0

h(u)du + Z <(a

)

<(a

)−δ

(<(a i ) − u)h(u)du

≥ β h (<(a i )) − 2δ

because khk 1 = 1 . Summing up all the terms

β h (<(a ^k ₁ )) + · · · + β h (<(a ^k _n )) ≥ β(<(a 1 )) + · · · + β (<(a n )) − 2nδ.

Then for every k ≥ K , we get

β(<(a ₁ )) + · · · + β(<(a _n )) ≤ 2nδ + β _h (<(a ^k ₁ )) + · · · + β _h (<(a ^k _n ))

≤ ε + N Φ

(s).

So

β(<(a 1 )) + · · · + β (<(a n )) ≤ ε + lim inf

k→+∞ N Φ

(s) and since n and ε are arbitrary, the result is proved.

Corollary 1. Let Φ a c ₀ -symbol with c ₀ ≥ 1 . Assume that sup

<(s)>0

N β,Φ (s)

β (<(s)) = C < +∞, then

sup

χ∈

sup

<(s)>0

N _β,Φ

(s) β(<(s)) = C.

Proof. For every τ ∈ R, it is easy to see that N β

,Φ

(s) = N β

,Φ (s + ic 0 τ ) and so for every τ ∈ R we have

sup

<(s)>0

N β,Φ

(s) β(<(s)) = C.

Now for χ ∈ T ^∞ , Φ χ is a vertical limit of a sequence (Φ τ

) _n≥0 where (τ n ) _n≥0 ⊂ R and then it suces to apply Proposition 3 to obtain that

sup

<(s)>0

N β,Φ

(s) β(<(s)) ≤ C

The other inequality is true because Φ = Φ χ with χ = (1, 1, . . . ) .

Remark. In the previous corollary, we can take the supremum on a band of the form {s ∈ C + , <(s) < θ} where θ ∈ R.

Let H a Hilbert space and K(H) the space of all compact operators on H . Let T be a bounded operator on H , we recall that its essential norm is dened by

kT k H,e = inf{kT − Kk, K ∈ K(H )}.

Clearly, the essential norm of T is zero if and only if T ∈ K(H ) . The next result

will be very useful.

Proposition. ([20],Prop5.1) Let T be a bounded operator on H and (K _n ) _n≥1 a sequence of self-adjoint compact operators on H . Let R _n = I − K _n . Assume that kR n k = 1 for every n ≥ 1 and that (R n ) _n≥1 converges pointwise to zero.

Then

kTk H,e = lim

n→+∞ kT R n k.

Theorem 3. Let Φ a c ₀ -symbol with c ₀ ≥ 1 . Assume that Im(ϕ) is bounded on C + , then

kC ϕ k _A

,e ≤

2 sup

C+

|Im(ϕ)| + c 0

· lim sup

<(s)→0

N β

,Φ (s) β h (Re(s)) and

kC ϕ k _H

,e ≤

2 sup

C+

|Im(ϕ)| + c 0

· lim sup

<(s)→0

N Φ (s)

<(s) . Corollary 2. Assume that Im(ϕ) is bounded on C + then:

(i) If N β

,Φ (s) = o(β(<(s)) when <(s) → 0 then C Φ is compact on A ² _µ . (ii) If N _Φ (s) = o(<(s)) when <(s) → 0 then C _Φ is compact on H ² . Remark. Cor2.(ii) is already known (see [3], Th2).

Proof of Th.3. We only give the proof in the case of A ² _µ , the proof in the case H ² is essentially the same.

For n ≥ 1 and f ∈ A ² _µ which has form (1) we dene



 

 



 

 



K n (f ) =

n

X

k=1

a k e k

R n (f ) = (I − K n )(f) =

+∞

X

k=n+1

a k e k .

Each K n is a self-adjoint compact operator on A ² _µ with norm 1 because K n

is a projection on a nite linear subspace and (R n ) clearly converges pointwise to zero. Then by the previous proposition

kC Φ k A

,e = lim

n→+∞ kC Φ R n k = lim

n→+∞

sup

kfk

≤1

kC Φ (R n (f ))k _A

.

In the sequel, we shall work on Ω = (0, +∞) × (0, 1) with the area measure: we apply the Littlewood-Paley formula for A ² _µ with the Lebesgue measure on (0, 1) :

kC _Φ (R _n (f ))k ² _A

µ

= |R _n (f ) ◦ Φ(+∞)| ² + 4

Z

T^∞

Z Z

Ω

β h (<(s))|(R n (f ) ◦ Φ)

_χ (s)| ² dsdm(χ).

We can point out that R n (f ) ◦ Φ(+∞) = 0 for every n ≥ 2 because c 0 ≥ 1

and then Φ(+∞) = +∞ . Now by Proposition 4.3 from [9]:

kC Φ (R n (f ))k ² _A

= 4

Z

T^∞

Z Z

Ω

β h (<(s))|R n (f )

_χ

(Φ χ (s))| ² |Φ

_χ (s)| ² dsdm(χ).

Now we make the following "change of variables": w = Φ _χ (s) , Φ _χ is not nec- essarily injective but we use the generalized formula of change of variable which involves the generalized Nevanlinna counting function (see [8] for example). We

get: Z Z

Ω

β h (<(s))|R n (f )

_χ

(Φ χ (s))| ² |Φ

_χ (s)| ² ds

= Z Z

Φ

(Ω)

|R n (f )

_χ

(s)| ² N β

,Φ

(s) ds.

Im(ϕ) is bounded then there exists A > 0 such that |Im(ϕ)| < A and the same inequality holds for ϕ _χ for every χ ∈ T ^∞ instead of ϕ : indeed ϕ _χ is a vertical limit of ϕ . Now, when s ∈ Ω , we have Im(s) ∈ (0, 1) and w = Φ _χ (s) veries Im(w) = c ₀ Im(s) + Im(ϕ _χ (s)) so Im(w) ∈ [−A, A + c ₀ ] . Finally if we denote Q = (0, +∞) × (−A, A + c 0 ) , we get

Z

Ω

β h (<(s))|R n (f )

_χ

(Φ χ (s))| ² |Φ

_χ (s)| ² ds

≤ Z Z

Q

|R n (f )

_χ

(s)| ² N β

,Φ

(s) ds.

Let us x θ > 0 , we write Q = Q _θ ∪ Q ˜ _θ where Q _θ = (0, θ) ×(−A, A +c ₀ ) and Q ˜ θ = (θ, +∞) × (−A, A + c 0 ) . We shall cut the previous integral in two parts.

Let γ θ dened by γ θ = sup

χ∈

sup

0<Re(s)<θ

N β

,Φ

(s) β _h (<(s))

= sup

0<Re(s)<θ

N β

,Φ (s) β _h (<(s)) . The second equality is true by Corollary 1. We have

Z

T^∞

Z Z

Q

|R n (f )

_χ

(s)| ² N _β

_,Φ

(s) dsdm(χ)

≤ γ _θ Z

T^∞

Z Z

Q

|R n (f )

_χ

(s)| ² β _h (<(s)) dsdm(χ).

Now we take the supremum:

sup

kfk

≤1

Z

T^∞

Z Z

Q

|R n (f)

_χ

(s)| ² N β

,Φ

(s) dsdm(χ)

≤ γ θ sup

kfk

≤1

Z

T^∞

Z Z

Q

|R n (f )

_χ

(s)| ² β h (<(s)) dsdm(χ)

≤ γ θ sup

kfk

≤1

Z

T^∞

Z Z

Q

|f _χ

(s)| ² β h (<(s)) dsdm(χ)

.

In the last inequality, we use the fact that if f belongs to the unit ball

of A ² _µ , R n (f ) too. Finally, by using again the Littlewood-Paley formula with

the normalized Lebesgue measure on (−A, A + c ₀ ) and the fact that the map χ → χ ^c

is measure preserving we obtain:

sup

kfk

≤1

4

Z

T^∞

Z Z

Q

|R _n (f )

_χ

(s)| ² N _β

_,Φ

(s) dsdm(χ)

≤ (2A + c ₀ )γ _θ . Now we obtain an upper bound for the second integral. By Proposition 2:

Z

T^∞

Z Z

Q ˜

|R _n (f )

_χ

(s)| ² N _β

_,Φ

(s) dsdm(χ)

≤ Z

T^∞

Z Z

Q ˜

|R _n (f )

_χ

(s)| ² β _h (<(s)) c 0

dsdm(χ) (∆)

Recall (see [3], lem2) that for every f ∈ H ² with form (1) and every proba- bility measure η on R we have for every σ > 0 :

Z

T^∞

Z

R

|f _χ

(σ + it)| ² dη(t)dm(χ) =

+∞

X

k=2

|a k | ² k ^−2σ log ² (k), ∀σ > 0.

Hence, we get:

(∆) = 2A + c 0

c 0

Z +∞

θ

+∞

X

k=n+1

β h (σ)|a k | ² k ^−2σ log ² (k)dσ.

Let us x ε > 0 . We claim that there exists K = K θ > 0 such that for every k ≥ K :

Z +∞

θ

β h (σ)k ^−2σ log ² (k)dσ ≤ εw h (k).

Indeed, we know by [16] that (w h (k)) is a decreasing sequence that decays more slowly than any negative power, so there exists C = C θ > 0 such that:

w h (k) ≥ Ck ^−θ ∀k ≥ 1.

Then Z +∞

θ

β _h (σ) k ^−2σ log ² (k) w h (k) dσ ≤

Z +∞

θ

σk ^θ−2σ C ⁻¹ log ² (k)dσ

because β h (σ) ≤ σ by denition of β h . Clearly the right-hand side of the in- equality goes to 0 when k goes to innity.

Finally when n ≥ K , we obtain 2A + c 0

c 0

Z +∞

θ

+∞

X

k=n+1

β h (σ)|a k | ² k ^−2σ log ² (k)dσ ≤ 2A + c 0

c 0

× ε

because we work with functions in A ² _µ with norm less than 1 . Now summing up the two integrals, we obtain that for n ≥ K ,

kC Φ R n k ≤ (2A + c 0 )γ θ + 2A + c 0

c 0

× ε ∀n ≥ K.

At last,

kC _Φ k _A

µ

,e = lim

n→+∞ kC _Φ R _n k

≤ (2A + c ₀ )γ _θ + 2A + c 0

c 0

× ε.

Now since ε and θ are arbitrary, we obtain the result.

B Necessary conditions of compactness

First, we recall some facts from [1]. The reproducing kernel at s ∈ C 1/2

associated to A ² _µ is dened for every w ∈ C 1/2 by

K µ,s (w) :=

+∞

X

n=1

n ^−s−w w _h (n) .

Example. Let α > −1 . The reproducing kernel at s ∈ C 1/2 associated to the space A ² _α is denoted by K _α,s instead of K _µ

_,s and we have for every w ∈ C 1/2 ,

K α,s (w) =

+∞

X

n=1

(1 + log(n)) ^α+1 n ^−s−w .

We saw in the previous section that the compactness is linked with the behaviour of Φ near to the imaginary axis. Then, as in [3], we shall work with partial reproducing kernels which are dened on C + and not only on C 1/2 . Denition. Let s ∈ C ⁺ and l ≥ 1 . On A ² _µ , we dene the partial reproducing kernel of order l at s ∈ C + by

K _µ,s ^l (w) := X

n≥1 p

(n)≤p

n ^−s−w w h (n) where p ⁺ (n) is the greatest prime divisor of n .

We claim that these partial reproducing kernels are dened on C + : indeed we know by [16] that the weight (w h (n)) decreases more slowly than any negative power of n so if ε > 0 , there exists C > 0 such that w h (n) > Cn ^−ε for every n ≥ 1 . Then

X

n≥1 p

(n)≤l

n −<(s)−<(w)

w h (n) ≤ 1 C

X

n≥1 p

(n)≤l

1

n <(s)+<(w)−ε = 1 C

l

Y

i=1

1−p −<(s)−<(w)+ε i

−1

.

For s ∈ C + , this quantity is then bounded for any w ∈ C ε and any ε > 0 and so the claim is proved.

We shall need the following proposition which follows easily from [3],Prop.5.

Proposition 4. Let Φ a c ₀ -symbol with c ₀ ≥ 1 and l ≥ 1 . Assume that Φ(s) = c ₀ s +

+∞

X

n=1

c _n n ^−s where c _n = 0 when p ⁺ (n) > l . Then C _Φ ^∗ (K _µ,s ^l ) = K _µ,Φ(s) ^l . Lemma 2. On the unit ball B _A

µ

of A ² _µ , the weak topology is the topology of

uniform convergence on every half-plane C 1/2+ε with ε > 0 .

Proof. Let (f _n ) ⊂ B _A

µ

which converges weakly to 0 . Let ε > 0 , by the result on the point evaluation from [1], there exists a constant C ε such that

|f n (s)| ≤ C ε kf n k for every s ∈ C 1/2+ε

and then the sequence (f n ) is uniformly bounded on C 1/2+ε . Now by the Mon- tel's theorem for H ^∞ (see [2],Lem18), there exists a subsequence which converges uniformly on C 1/2+ε . It suces to show now that 0 is the unique cluster point, but the point evaluation is bounded and (f n ) converges weakly to 0 and so 0 is the unique cluster point.

Conversely let (f n ) ⊂ B _A

which converges uniformly to 0 on C 1/2+ε for every ε > 0 . There exists at least a subsequence which converges weakly to f ∈ A ² _µ . But the point evaluation is bounded for every s ∈ C 1/2 and (f n ) converges uniformly to 0 on C 1/2+ε for every ε > 0 so f = 0 .

Theorem 4. Let Φ a c 0 -symbol with c 0 ≥ 1 and l ≥ 1 . Assume that Φ(s) = c 0 s +

+∞

X

n=1

c n n ^−s where c n = 0 when p ⁺ (n) > l . We have:

kC Φ k _A

,e ≥ lim sup

<(s)→0

kK _µ,Φ(s) ^l k _A

kK _µ,s ^l k _A

.

Proof. Let l ≥ 1 . For s ∈ C + , we dene the following family of functions:

k s (w) := K _µ,s ^l (w) kK _µ,s ^l k _A

, where w ∈ C + . This family is contained in B _A

µ

and converges uniformly to 0 on C 1/2+ε for every ε > 0 : indeed we have

lim

<(s)→0 kK _µ,s ^l k ² _A

µ

= lim

<(s)→0 K _µ,s ^l (s) = +∞

and for every w ∈ C 1/2+ε :

|K _µ,s ^l (w)| ≤ X

n≥1 p

(n)≤l

n ^−1/2−ε w h (n) < +∞

because the partial reproducing kernels are dened on C + .

Let K ∈ K(A ² _µ ) , we point out that K ^∗ ∈ K(A ² _µ ) as well and by the previous lemma we know that (k s ) converges weakly to 0 so K ^∗ (k s ) is norm convergent to 0 . Then

kC Φ − Kk = kC _Φ ^∗ − K ^∗ k ≥ kC _Φ ^∗ (k s ) − K ^∗ (k s )k _A

≥ kC _Φ ^∗ (k s )k _A

− kK ^∗ (k s )k _A

and then

kC Φ − Kk ≥ lim sup

<(s)→0

kC _Φ ^∗ (k _s )k _A

.

Now with Proposition 4, and the fact that the inequality is true for every K ∈

K(A ² _µ ) , we obtain the result.

Corollary 3. Let Φ a c ₀ -symbol with c ₀ ≥ 1 and l ≥ 1 . Assume that Φ(s) = c ₀ s +

+∞

X

n=1

c _n n ^−s where c _n = 0 when p ⁺ (n) > l . Assume that C _Φ ∈ K(A ² _µ ) , then:

lim

<(s)→0

kK _µ,Φ(s) ^l k _A

kK _µ,s ^l k _A

= 0.

In particular if C Φ ∈ A ² _α with α > −1 , it implies that lim

<(s)→0

<(s)

<(Φ(s)) = 0.

Remark. The result concerning A ² _α is already known (see [3]).

C A criterion of compactness

In this section, we still work on A ² _µ where µ is a probability measure such that dµ(σ) = hdσ with h a positive continuous function. Recall that

β h (σ) = Z σ

0

(σ − u)h(u)du.

In the spirit of [11] (see p12) we introduce a new condition: we say that a weight β veries condition (κ) if the function G = G β dened by G(σ) = β(σ)/σ (where σ > 0 ) is such that

η→0 lim

lim sup

σ→0

G(ησ) G(σ) = 0.

Examples.

(i) When h is a non decreasing function then (κ) is fulllled for β h : indeed in this case we extend continuously G at 0 by G(0) = 0 and it suces to check that G is a convex function but it is easy to see that

G ⁰ (σ) = Z σ

0

uh(u)du σ ² and

G ⁰⁰ (σ) = 2

Z σ 0

u(h(σ) − h(u))du

σ ³ .

(ii) Let α > −1 , β _α veries (κ) because G(σ) ≈ σ ^α+1 when σ is small enough.

Proposition 5. Let Φ a c 0 -symbol with c 0 ≥ 1 . Assume that β h veries (κ) then if

lim

<(s)→0

<(s)

<(Φ(s)) = 0 then

N β

,Φ (s) = o(β h (<(s))) when <(s) → 0,

hence C _Φ is compact on A ² _µ if Im(ϕ) is bounded.

Proof. Let s ∈ C + , we have

N β

,Φ (s) = X

a∈

Φ(a)=s

β h (<(a))

= X

a∈

Φ(a)=s

G(<(a))<(a).

Let ε > 0 , we know that

<(s)→0 lim

<(s)

<(Φ(s)) = 0 and then by Condition (κ) , there exists δ > 0 such that

G(<(s)) ≤ εG(<(Φ(s)))

when <(s) < δ . Now, thanks to the Schwarz's lemma if Φ(a) = s and <(s) < δ then <(a) < δ too. Consequently

N β

,Φ (s) = X

a∈

Φ(a)=s

G(<(a))<(a)

≤ ε X

a∈

Φ(a)=s

G(<(Φ(a)))<(a)

= εG(<(s)) X

a∈C

Φ(a)=s

<(a)

= εG(<(s))N Φ (s)

for every s ∈ C + such that <(s) < δ . Now by Littlewood's inequality we get:

N β

,Φ (s) ≤ εG(<(s))<(s) c 0

= εβ _h (<(s)) c 0

when <(s) < δ and this proves the result.

Theorem 5. Let α > −1 , Φ a c ₀ -symbol with c ₀ ≥ 1 and l ≥ 1 . Assume that Φ(s) = c 0 s +

+∞

X

n=1

c n n ^−s where c n = 0 when p ⁺ (n) > l and that Im(ϕ) is bounded, then the following assertions are equivalent:

(i) C _Φ is compact on A ² _α . (ii) lim

<(s)→0

<(s)

<(Φ(s)) = 0.

(iii) N _α,Φ (s) = o((<(s)) ^α+2 ) when <(s) goes to 0 .

Proof. It suces to use Corollary 3, Proposition 5 and Corollary 2.

Remark. We point out that the condition (ii) does not depend on α .

Denition. Let Φ : C + → C + be an analytic function and k ∈ N, we say that

Φ is k− valent if for every w ∈ C ⁺ , there exists at most k solutions to Φ(s) = w .

If Φ is k− valent for some k ∈ N then we shall say that Φ is nite-valent.

Theorem 6. Let l ≥ 1 and Φ be a c ₀ -symbol with c ₀ ≥ 1 such that Φ(s) = c ₀ s + P +∞

n=1 c _n n ^−s where c _n = 0 if p ⁺ (n) > l . Assume that Im(ϕ) is bounded and that Φ is nite-valent, then the following assertions are equivalent:

(i) C Φ is compact on H ² . (ii) lim

<(s)→0

<(s)

<(Φ(s)) = 0.

(iii) N _Φ (s) = o(<(s)) when <(s) goes to 0 .

Proof. We only have to proof that (ii) ⇒ (iii) : the other implications have been proven in [3]. We assume that Φ is k -valent for some k ≥ 0 . Let ε > 0 and s ∈ C + , as in the proof of Proposition 5 we can show that there exists δ > 0 such that for every ∈ C + with <(s) < δ :

N _Φ (s) ≤ ε<(s) X

a∈

Φ(a)=s

1 ≤ εk<(s).

Remark. Let l ≥ 1 and Φ a c ₀ -symbol with c ₀ ≥ 1 such that Φ(s) = c ₀ s + P +∞

n=1 c n n ^−s where c n = 0 if p ⁺ (n) > l . If Im(ϕ) is bounded and C ϕ is compact on H ² then

lim

<(s)→0

<(s)

<(Φ(s)) = 0 and consequently C Φ is compact on A ² _α for any α > −1 .

Questions. Is the previous implication true for general c ₀ -symbols?

5 Compactness and Carleson measures

In this section, we are going to obtain a sucient condition of compactness for C Φ with a "Carleson-measure" condition. We still work on A ² _µ where µ is a probability measure on (0, +∞) such that 0 ∈ Supp(µ) .

When Φ is a c ₀ -symbol with c ₀ ≥ 1 , it is shown in [2] that Φ admits radial limits and so if λ is the Lebesgue measure on R,

Φ ^∗ (it) := lim

σ→0

Φ(σ + it) exists λ -almost everywhere.

Notation. Let t ∈ R and h > 0 . We dene the Carleson window centered at it and of size h by H (t, h) := {s ∈ C + , |s − it| < h} .

Denition 2.

(i) We denote λ Φ the pullback measure of λ by Φ ^∗ , ie, for every open set Ω ⊂ C + :

λ Φ (Ω) := λ({t ∈ R , Φ ^∗ (it) ∈ Ω}).

The Carleson function associated to λ _Φ is then dened by ρ Φ (h) := sup

t∈

λ Φ (H(t, h)).

(ii) Let λ _µ = λ ⊗ µ . We denote by λ _µ,Φ the pullback measure of λ _µ by Φ , ie, for every open set Ω ⊂ C + :

λ µ,Φ (Ω) := λ µ ({s ∈ C + , Φ(s) ∈ Ω}).

The Carleson function associated to λ Φ is then dened by ρ µ,Φ (h) := sup

t∈R

λ µ,Φ (H(t, h)).

In the sequel we shall denote β instead of β h . The next theorem is the main result of this section.

Theorem 7. Let Φ be a c 0 -symbol with c 0 ≥ 1 . There exists K > 0 (independent of Φ ) such that:

(i)

sup

s∈H(t,h/2)

N _Φ (s) ≤ K · λ _Φ H(t, 2c ₀ h) for every h small enough.

In particular:

s∈ sup

<(s)<h/2

N Φ (s) ≤ K · ρ Φ 2c 0 h

for every h small enough.

(ii)

sup

s∈H(t,h/2)

N β,Φ (s) ≤ K · λ µ,Φ H (t, 2c 0 h)

for every h small enough.

In particular:

sup

s∈C

<(s)<h/2

N β,Φ (s) ≤ K · ρ µ,Φ 2c 0 h for every h small enough.

Corollary 4. Let Φ be a c ₀ -symbol with c ₀ ≥ 1 such that Im(ϕ) is bounded on C + . Then there exists C > 0 such that:

(i)

kC Φ k _H

,e ≤ C lim sup

h→0

ρ _Φ (h) h .

In particular if ρ Φ (h) = o(h) when h → 0 then C Φ is compact on H ² . (ii)

kC Φ k _A

µ

,e ≤ C lim sup

h→0

ρ µ,Φ (h) β(h) .

In particular if ρ µ,Φ (h) = o(β(h)) when h → 0 then C Φ is compact on A ² _µ . Example. When µ = µ α , we denote N α (resp. ρ α,Φ ) the corresponding gen- eralized Nevanlinna counting function (resp. the Carleson function). We know that in that case, β _α (σ) ≈ σ ^α+2 when σ is small and then if Im(ϕ) is bounded:

kC Φ k _e,A

α

≤ C lim sup

h→0

ρ α,Φ (h)

h ^α+2 .

Proof of Cor.4. We only give the proof of (i) . Let s ∈ C + with <(s) small enough, we have

N Φ (s) ≤ sup

s

∈H(Im(s),2<(s))

N Φ (s ⁰ ) ≤ K · ρ Φ 8c 0 <(s) by theorem 7. Then we obtain

N _Φ (s)

<(s) ≤ 8c 0 K · ρ Φ 8c 0 <(s) 8c 0 <(s) and so the result.

In order to prove Theorem 7, we need to recall some facts about Carleson measures and the classical Nevanlinna counting function on the unit disk.

Let φ : D → D be an analytic self-map. The Nevanlinna counting function of φ is dened for w ∈ φ( D ) by

N _φ (w) = X

z∈

φ(z)=w

log(1/|z|).

When w / ∈ φ( D ) , N _φ (w) = 0 by convention.

We denote by m _φ the pullback measure of the Lebesgue measure on the unit circle by φ ^∗ , the boundary values function of φ (which exists because φ ∈ H ^∞ ( D ) ). By abusing the notation, we still denote φ instead of φ ^∗ .

For η ∈ T and h > 0 , the Carleson window S(η, h) is dened by S(η, h) := {z ∈ D , |z − η| < h}.

We shall make a crucial use of the following result (see [14],Th1.1):

Let φ : D → D an analytic self-map. For every β > 1 , there exists a universal constant C _β > 0 such that:

(1/C β )m φ (S(ξ, h)) ≤ sup

z∈S(ξ,h)

N φ (z) ≤ C β m φ (S(ξ, βh)) ~ ~ for every 0 < h < (1 − |φ(0)|)/β and every ξ ∈ T.

Proof of Th.7(i).

B Step one: We prove the result when t = 0 .

We shall use a family of functions (ψ ξ ) ξ>0 where ψ ξ is dened by







ψ _ξ : C + −→ D s 7−→ s − ξ

s + ξ . Claim. For every h > 0 , H (0, h/2) ⊂ ψ _c ⁻¹

0

ξ (S(−1, h/ξ)) . Let us prove this fact:

s ∈ ψ ⁻¹ _c

0

ξ (S(−1, h/ξ)) ⇔ ψ _c

_ξ (s) ∈ S(−1, h/ξ) ⇔

s − c ₀ ξ s + c 0 ξ + 1

< h ξ

⇔

2s s + c 0 ξ

< h

ξ ⇔ |s| < h 2

s + c 0 ξ ξ

.

Now if s ∈ H (0, h/2) then |s| < ^h ₂ and it suces to remark that for every ξ > 0 ,

|s + c 0 ξ|

ξ ≥ <(s + c 0 ξ)

ξ ≥ c 0 ≥ 1, and this justies the claim.

Let h > 0 and s ∈ H (0, h/2) . Then w ξ := ψ c

ξ (s) ∈ S(−1, h/ξ) . As- sume that s ∈ Φ( C + ) (if it is not the case, there is nothing to prove) and let s 1 , . . . , s n ∈ C + such that Φ(s i ) = s for every i = 1, . . . , n .

For ξ > 0 , we dene the analytic self-map Θ _ξ : D → D by Θ _ξ = ψ _c

_ξ ◦Φ◦ψ _ξ ⁻¹ . We note that Θ _ξ (ψ _ξ (s _i )) = ψ _c

_ξ (Φ(s _i )) = ψ _c

_ξ (s) = w _ξ and so

n

X

i=1

log(1/|ψ ξ (s i )|) ≤ N Θ

(w ξ ).

For i ∈ {1, . . . , n} we have log(1/|ψ ξ (s i )|) = 1

2 log

|s i | ² + 2ξ<(s i ) + ξ ²

|s i | ² − 2ξ<(s i ) + ξ ²

= 1 2 log

1 + 4ξ<(s i )

|s i | ² − 2ξ<(s i ) + ξ ² . Let ε > 0 . For ξ large enough we obtain that for every i ∈ {1, . . . , n} :

2(1 − ε) <(s i )

ξ ≤ log(1/|ψ ξ (s i )|).

And so

2(1 − ε)

n

X

i=1

<(s i )

ξ ≤ N Θ

(w ξ ).

Now using ~~ with β = 2 and remembering that w _ξ ∈ S (−1, h/ξ) for every ξ > 0 , we obtain

N Θ

(w ξ ) ≤ C 2 m Θ

(S(−1, 2h/ξ)) for every 0 < h/ξ < (1 − |Θ ξ (0)|)/2 . Point out that

Θ ξ (0) = ψ c

ξ (Φ(ξ)) = ϕ(ξ) 2c 0 ξ + ϕ(ξ)

and this quantity goes to 0 when ξ goes to innity. So we dene m 0 = inf

ξ>1 ξ(1 − |Θ ξ (0)|)/2 > 0.

It is clear that m 0 ≤ 1/2 . Let us point out that m 0 = m 0 (ϕ) = m 0 (ϕ(· + iτ)) for any τ ∈ R (this remark will be useful in step 2 ). When h < m 0 , we get

N Θ

(w ξ ) ≤ C 2 m Θ

(S(−1, 2h/ξ)) and

m _Θ

(S(−1, 2h/ξ)) = m({η ∈ T , Θ _ξ (η) ∈ S(−1, 2h/ξ)})

= m({η ∈ T , Ψ c

ξ ◦ Φ ^∗ ◦ ψ _ξ ⁻¹ (η) ∈ S(−1, 2h/ξ)})

= Z

{Ψ

◦Φ

◦ψ

(e

)∈S(−1,2h/ξ)})

dθ

2π .

Now we make the following change of variables iu = ψ ⁻¹ _ξ (e ^iθ ) : m Θ

(S(−1, 2h/ξ)) =

Z

{Φ

(iu)∈Ψ

0ξ

(S(−1,2h/ξ))}

ξ π(ξ ² + u ² ) du

= Z

{|Φ

(iu)|<h

}

ξ

π(ξ ² + u ² ) du.

Then for ξ > 1 large enough we obtain 2(1 − ε)

n

X

i=1

Re(s _i ) ξ ≤ C ₂

Z

{|Φ

(iu)|<h

}

ξ π(ξ ² + u ² ) du and then

2(1 − ε)

n

X

i=1

<(s i ) ≤ C 2

Z

{|Φ

(iu)|<h

}

du π .

But if |Φ ^∗ (iu)| < h ^|Φ

^(iu)+c _ξ

^ξ| then |Φ ^∗ (iu)| < _(1−h/ξ) ^c

^h < _(1−h) ^c

^h < 2c 0 h since h < m 0 ≤ 1/2 . Then

2(1 − ε)

n

X

i=1

Re(s i ) ≤ C 2 m Φ (H(0, 2c 0 h)).

Since n is arbitrary we get:

2(1 − ε)N _Φ (s) ≤ C ₂ m _Φ (H (0, 2c ₀ h)).

for any ε > 0 and the rst step is proved.

B Step two: we prove now the result when t 6= 0 . We have

sup

s∈H(c

t,h/2)

N _Φ (s) = sup

s∈H(0,h/2)

N _Φ (s + ic ₀ t).

But if Φ(s) := ˜ c ₀ s + ϕ(s + it) = Φ(s + it) − ic _o t , we obtain for s ∈ H(0, h/2) : N Φ (s + ic 0 t) = X

a∈C

Φ(a)=s+ic

t

<(a) = X

a∈C

c

(a−it)+ϕ(it+a−it)=s

<(a − it)

= X

a

∈

Φ(a ˜

)=s

<(a ⁰ ) = N Φ ˜ (s).

Applying step one to Φ ˜ , we get sup

s∈H(0,h/2)

N Φ (s + ic 0 t) = sup

s∈H(0,h/2)

N Φ ˜ (s) ≤ Kλ Φ ˜ (H (0, 2c 0 h)) for every h small enough. Now it suces to remark that

λ Φ ˜ (H (0, 2c 0 h)) = λ({t ⁰ ∈ R , Φ ˜ ^∗ (it ⁰ ) ∈ H (0, 2c 0 h)})

= λ({t ⁰ ∈ R , |Φ ^∗ (it + it ⁰ ) − ic 0 t| < 2c 0 h})

= λ({t ⁰⁰ ∈ R , |Φ ^∗ (it ⁰⁰ ) − ic 0 t| < 2c 0 h})

= λ Φ (H (c 0 t, 2c 0 h)).

Proof of Th.7(ii). By Lemma 1, we know that N β

,Φ (s) =

Z <(s) 0

N Φ

(s)h(u)du.

Now in the proof of (i) , we point out that we can choose the same m ₀ for Φ _u for every u ≥ 0 and so for s ∈ H (t, h/2) ˜ with ˜ h < m 0 we have

N β

,Φ (s) ≤ K Z <(s)

0

λ Φ

(H (t, 2c 0 h))h(u)du ˜

≤ K Z +∞

0

λ({t ⁰ ∈ R , Φ(u + it ⁰ ) ∈ H (t, 2c 0 ˜ h)})h(u)du = Kλ µ,Φ (H(t, 2c 0 ˜ h) thanks to the Fubini's theorem.

Acknowledgements. I would like to thank Pascal Lefèvre for his advice regarding this paper.

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Univ Lille-Nord-de-France UArtois,

Laboratoire de Mathématiques de Lens EA 2462, Fédération CNRS Nord-Pas-de-Calais FR 2956, F-62 300 LENS, FRANCE

[email protected]